Arithmetic of fundamental groups
Abstract. Étale fundamental groups unite classical Galois groups
with (profinitely completed) topological fundamental groups. For a
variety X/k, this leads to an (outer) action of the absolute
Galois group Galk on a profinitely completed
topological fundamental
group. The study of this arithmetic structure on fundamental groups
is the central objective of anabelian geometry.
The lectures will focus on the section conjecture of Grothendieck:
the prediction that the set of rational points for smooth, projective
curves of genus at least 2 over number fields is in bijection with
conjugacy classes of sections of the fundamental exact sequence.
Topics to be covered in the lectures are: foundations for the
profinite Kummer map, cycle classes of sections, the section
conjecture over various local fields and even finite fields,
and local-to-global aspects of the section conjecture.